Centripetal Acceleration Change with Speed and Radius

Centripetal acceleration is the acceleration directed towards the center of a circular path that causes an object to move in a circle. Its magnitude depends on the speed of the object and the radius of the circular path.

The formula for the magnitude of centripetal acceleration (\(a_c\)) is given by:

$$a_c = \frac{v^2}{r}$$

where:

• \(v\) is the speed of the body
• \(r\) is the radius of the circular path

The question states that the speed (\(v\)) and the radius (\(r\)) of the body moving in a circular path are doubled. Let's denote the new speed as \(v'\) and the new radius as \(r'\).

• New speed, \(v' = 2v\)
• New radius, \(r' = 2r\)

Now, let's find the new magnitude of centripetal acceleration, \(a_c'\), using the formula with the new speed and new radius:

$$a_c' = \frac{(v')^2}{r'}$$

Substitute the new values \(v' = 2v\) and \(r' = 2r\) into the formula:

$$a_c' = \frac{(2v)^2}{2r}$$

Simplify the expression:

$$a_c' = \frac{4v^2}{2r}$$

$$a_c' = 2 \times \frac{v^2}{r}$$

We know that the original centripetal acceleration is \(a_c = \frac{v^2}{r}\). So, we can substitute \(a_c\) into the equation for \(a_c'\):

$$a_c' = 2 \times a_c$$

This shows that the new magnitude of centripetal acceleration (\(a_c'\)) is twice the original magnitude of centripetal acceleration (\(a_c\)). Therefore, the magnitude of centripetal acceleration will be doubled.

Let's check the options provided:

1. half
2. four times
3. doubled
4. More than one of the above
5. None of the above

Our calculation shows the centripetal acceleration is doubled, which matches option 3.